fermi_surface/theory.md

@@ -22,22 +22,21 @@ computations with the PKKprime code.
 
 The goal is to solve numerically the secular equation of KKR :
 
-$`\|\underline{\underline{M}}(\vec{k},E)\|=0`$ where
-$`\underline{\underline{M}}(\vec{k},E)=\delta_{\Lambda\Lambda\'}-\sum\limits_{\Lambda*}
-g_{\Lambda\Lambda*}(\vec{k};E) t_{\Lambda\'\'\Lambda\'}(E)`$ and
-$`\ \Lambda=(L,\sigma)`$.
+$`|\underline{\underline{M}}(\vec{k},E)|=0`$ where
+$`\underline{\underline{M}}(\vec{k},E)=\delta_{\Lambda\Lambda'}-\sum\limits_{\Lambda*}
+g_{\Lambda\Lambda*}(\vec{k};E) t_{\Lambda''\Lambda'}(E)`$ and
+$`\Lambda=(L,\sigma)`$.
 
 In most cases this equation can\'t be solved analytically so a "trick"
 is done. This trick consists on rather solving:
 
-$`\underline{\underline{M}}(\vec{k},E) \underline{c_\nu} =
-\lambda_\nu(\vec{k},E) \underline{c_\nu}`$
+$`\underline{\underline{M}}(\vec{k},E) \underline{c_\nu} = \lambda_\nu(\vec{k},E) \underline{c_\nu}`$
  and find the
 eigenvalue which satisfies the following condition
 $`\lambda_\nu(\vec{k},E)=0`$.
  Thus the objective is to scan the
 k-space, for a fixed energy $`E`$, and find
-$`\min\limits_{\vec{k},\nu}(\|\lambda_\nu(\vec{k},E)\|)`$.
+$`\min\limits_{\vec{k},\nu}(|\lambda_\nu(\vec{k},E)|)`$.
 
 ## Qualitative explanation of the iterative method
 
@@ -51,14 +50,14 @@ scanning the k-space can be a heavy computational task, especially for
 refine the mesh and keep in memory only the zones crossed by a band. So
 in a first step the algorithm will quad the Brillouin zone, into
 tetrahedras (triangles in 2D), and then evaluate if along the edges we
-can find a solution satisfying $`\|\lambda_\nu(\vec{k},E)\|\<a_n`$,
+can find a solution satisfying $`|\lambda_\nu(\vec{k},E)|\<a_n`$,
 where $`a_n`$ is a given accuracy.
 
 If a solution is found along a tetrahedra edges, the mesh is refined in
-this region. Otherwise, we don\'t take any more into account this
+this region. Otherwise, we don't take any more into account this
 spatial region. This step is repeated until we reach the final number of
-desired cubes in the Brillouin zone ((Total cubes number = cubes with
-intersect the bands and cubes which doesn\'t intersect the bands)).
+desired cubes in the Brillouin zone (Total cubes number = cubes with
+intersect the bands and cubes which doesn't intersect the bands).
 
 Once the mesh is refined to a certain accuracy, the research of
 $`\vec{k}`$ that minimize $`\lambda_\nu(\vec{k},E)`$ start in the
@@ -66,7 +65,7 @@ cubes which are crossed by a band (found in the previous step). For
 doing so a dichotomy procedure is employed:
 
  - The spectrum of $`\underline{\underline{M}}(\vec{k},E)`$ is calculated for discrete k-points in the tetrahedra edges.
-- The local $`\Omega_k`$ region where the transition $`\lambda_\nu(\vec{k},E) > 0 \rightarrow \lambda_\nu(\vec{k},E) < 0`$ is kept in memory. So by the Bijection Theorem,  $`\exists \vec{k}_0 \in \Omega_k,  \lambda_\nu(\vec{k}_0,E)=0`$.
+ - The local $`\Omega_k`$ region where the transition $`\lambda_\nu(\vec{k},E) > 0 \rightarrow \lambda_\nu(\vec{k},E) < 0`$ is kept in memory. So by the Bijection Theorem,  $`\exists \vec{k}_0 \in \Omega_k, \lambda_\nu(\vec{k}_0,E)=0`$.
  - The k-mesh in $`\Omega_k`$ is refined.
  - The same steps are repeated until the given accuracy is reached, or the maximal number of steps is exhausted.